Limitations of the Invertible-Map Equivalences

TL;DR

This paper demonstrates fundamental limitations of invertible-map equivalences in distinguishing certain graph structures, showing they cannot fully characterize graph isomorphism or capture polynomial time in fixed-point logic with linear algebra.

Contribution

It unifies previous results to prove that invertible-map equivalences over all primes cannot fully identify graph isomorphism or characterize polynomial time.

Findings

Invertible-map equivalences do not coincide with graph isomorphism.

Such equivalences cannot capture polynomial time.

Limits of linear-algebraic logic in graph isomorphism testing.

Abstract

This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Gr\"adel, and Wied Pakusa, published at ICALP 2019 and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences $\equiv^\text{IM}_{k,Q}$ on certain variants of Cai-F\"urer-Immerman (CFI) structures. These $\equiv^\text{IM}_{k,Q}$-equivalences, for a number $k$ and a set of primes~$Q$, refine the well-known Weisfeiler-Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs $G\equiv^\text{IM}_{k,Q}H$ cannot be distinguished by iterative refinements of equivalences on $k$-tuples defined via linear operators on vector spaces over fields of characteristic $p \in Q$. In the first paper it has been shown, using considerable algebraic machinery, that for a prime $q \notin Q$, the $\equiv^\text{IM}_{k,Q}$ equivalences do not distinguish between non-isomorphic CFI-structures over the field $\mathbb{F}_q$. In the second paper, a similar but not identical construction for CFI-structures over the rings $\mathbb{Z}_{2^i}$ has, again by rather involved combinatorial and algebraic arguments, been shown to be indistinguishable with respect to $\equiv^\text{IM}_{k,\{2\}}$. Together with earlier work, this second result separates rank logic from polynomial time. We unify the two approaches and prove that CFI-structures over the rings $\mathbb{Z}_{2^i}$ are indistinguishable with respect to $\equiv^\text{IM}_{k,\mathbb{P}}$, for the set $\mathbb{P}$ of all primes. This implies the following two results. 1. There is no fixed $k$ such that the invertible-map equivalence $\equiv^\text{IM}_{k,\mathbb{P}}$ coincides with isomorphism on all finite graphs. 2. No extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time.